Abstract

In this paper, we consider the existence of positive solutions for a discrete third-order boundary value problems, which has the sign-changing Green’s function. The approach we use is the Guo-Krasnoselskii fixed point theorem in a cone.

Keywords

1 Introduction

Let a, b be two integers with \(b>a\). Let us employ \([a,b]_{\mathbb{Z}}\) to denote the integer set \(\{a,a+1,\ldots, b\}\). For any real number \(c>1\), \([c]\) is the integer part of c. In this paper, we consider the existence of a positive solution for the following discrete third-order BVP:

where \(T>3\) is an integer, \(\eta\in [1, [\frac{3T^{2}-3T-2}{6T+3} ] ]_{\mathbb{Z}}\), \(a:[1,T-1]_{\mathbb{Z}}\to(0,+\infty)\) and \(f:[1,T-1]_{\mathbb{Z}}\times[0,+\infty)\to[0,+\infty)\) is continuous.

Difference equations appear in many mathematical models in diverse fields, such as economy, biology, physics, and finance; see [1–3]. In recent years, the existence and multiplicity of positive solutions of discrete boundary value problems have received much attention from many authors and a great deal of work has been done by using classical methods such as fixed point theory [4–8], lower and upper solutions methods [9], critical point theory [10–12], etc.

However, in all of the above papers, in order to obtain positive solution, the Green’s functions they used are positive. Now, there is a question: when the Green’s function changes its sign, can we get the existence of a positive solution?

In this paper, we will consider the existence of a positive solution of (1.1). It will be shown that the Green’s function of (1.1) changes its sign in Section 2.

Finally, it must be mentioned that there are some excellent results on the existence of the positive solutions of BVPs for ordinary differential equations when the Green’s functions change their signs; see Sun et al. [19–21] and Palamides et al. [22, 23] and references therein.

Our main tool is the following well-known Guo-Krasnoselskii fixed point theorem.

ThenAhas at least one fixed point in\(K\cap(\overline {\Omega}_{2}\backslash\Omega_{1})\).

The rest of this paper is arranged as follows. In Section 2, we will show the expression and some properties of the Green’s function of (1.1). Specially, we will show that the Green’s function changes its sign. Moreover, we will give some other preliminaries. In Section 3, we will demonstrate our main result and prove it.

Now, we will show that if \(\eta\in [1, [\frac {3T^{2}-3T-2}{6T+3} ] ]_{\mathbb{Z}}\), then \(u(t)\geq 0\) for \(t\in[0,T+1]_{\mathbb{Z}}\).

Let \(\phi(t)=t^{2}-3(1+\eta)t-3T(T-1)+3(2T+1)\eta+2\). It is obvious that \(u(t)\geq0\) equals \(\phi(t)\leq0\). Since \(\Delta\phi(t)=2t-3\eta-2\), we get \(\Delta\phi(t)\geq0\) for \(t\geq\frac{3\eta}{2}+1\) and \(\Delta\phi(t)\leq0\) for \(t\leq \frac{3\eta}{2}+1\). Due to \(\frac{2T}{3}>\frac{3T^{2}-3T-2}{6T+3}\), we get \(\frac{3\eta}{2}+1< T+1\). Therefore, \(\Delta\phi(t)\geq0\) for \(t\in [ [\frac{3\eta}{2}+1 ],T ]_{\mathbb{Z}}\) and \(\Delta\phi(t)\leq0\) for \(t\in [0, [\frac{3\eta }{2}+1 ] ]_{\mathbb{Z}}\), i.e., \(\phi(t)\) is increasing on \([ [\frac{3\eta}{2}+1 ],T+1 ]_{\mathbb{Z}}\) and \(\phi(t)\) is decreasing on \([0, [\frac{3\eta}{2}+1 ] ]_{\mathbb{Z}}\). Consequently, if \(\phi(0)\leq 0\) and \(\phi(T+1)\leq0\), then \(\phi(t)\leq0\) for \(t\in [0,T+1]_{\mathbb{Z}}\). By the direct computation, \(\phi(0)\leq0\) for \(\eta\in [1, [\frac{3T^{2}-3T-2}{6T+3} ] ]_{\mathbb{Z}}\) and \(\phi(T+1)\leq0\) for \(\eta\in [1, [\frac{2T^{2}-2T}{3T} ] ]_{\mathbb{Z}}\). Combining with the fact \(\frac{3T^{2}-3T-2}{6T+3}<\frac{2T^{2}-2T}{3T}<\frac{2T}{3}\), we get \(\eta\in [1, [\frac{3T^{2}-3T-2}{6T+3} ] ]_{\mathbb{Z}}\).

Now, let us give some notations.

Let \(E=\{u:[0,T+1]_{\mathbb{Z}}\to\mathbb{R}\}\). Then E is a Banach space under the norm \(\|u\|=\max_{t\in[0,T+1]_{\mathbb{Z}}}|u(t)| \). Let

Since \(\Delta u(t)\geq0\) for \(t\in[0,T]_{\mathbb{Z}}\), we have \(u(t)\geq0\) for \(t\in[0,T+1]_{\mathbb{Z}}\). So, \(u\in K_{0}\). Due to \(\Delta^{2}u(\eta)=0\), we have \(\Delta^{2}u(t-1)\leq0\) for \(t\in [\eta+1,T]_{\mathbb{Z}}\), i.e., \(u(t)\) is concave on \([\eta +1,T+1]_{\mathbb{Z}}\). □

This shows that \(\|Au\|\leq\|u\|\), \(u\in K\cap\partial\Omega_{1}\).

Similarly, for any \(u\in K\cap\partial\Omega_{2}\), we get \(\theta ^{\ast}R\leq u(s)\leq R\) for \(s\in[\theta,T+1-\theta]_{\mathbb{Z}}\) by Lemma 2.3. Since the function \(G(t,s)\) is positive and increasing for \(\eta< s\leq T-1\), it follows from the assumption (A2) that

This indicates that \(\|Au\|\geq\|u\|\), \(u\in K\cap\partial\Omega_{2}\).

Therefore, A has a fixed point \(u\in K\cap(\overline{\Omega} _{2}\backslash\Omega_{1})\) from Theorem 1.1, which is a positive and increasing solution of the BVP (1.1) with \(r\leq\|u\|\leq R\). Moreover, we know the obtained solution u is concave on \([\eta +1,T+1]_{\mathbb{Z}}\) from the proof of Lemma 2.2.

Therefore, by Theorem 1.1, we obtain a solution of the problem (1.1). Moreover, we know the obtained solution u is concave on \([\eta +1,T+1]_{\mathbb{Z}}\) from the proof of Lemma 2.2. So, the proof of Corollary 3.1 is completed. □

Declarations

Acknowledgements

The authors are very grateful to the anonymous referees for their valuable suggestions. The authors are supported by NWNU-LKQN-11-23, NSFC (11401479, 11201378), Natural Science Foundation of Gansu Province (145RJYA237), The Science Research Project for Colleges and Universities of Gansu Province (2013A-001), China Postdoctoral Science Foundation (2014M562472).

Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.